question
find the average rate of change of the function
$f(x)$, given below, from $x = 3$ to $x = t$.
$f(x)=4x^{2}+x - 2$
give your answer in terms of $t$.
provide your answer below:

Question

Accepted Answer

# Answer:
$4t + 13$
# Explanation:
## Step1: Recall the formula for average rate of change
The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 3$ and $b=t$, and $f(x)=4x^{2}+x - 2$.
## Step2: Calculate $f(3)$
Substitute $x = 3$ into $f(x)$:
$$
\begin{align*}
f(3)&=4\times(3)^{2}+3- 2\
&=4\times9+3 - 2\
&=36+3-2\
&=37
\end{align*}
$$
## Step3: Calculate $f(t)$
Substitute $x = t$ into $f(x)$:
$$
f(t)=4t^{2}+t - 2
$$
## Step4: Calculate the average rate of change
$$
\begin{align*}
\frac{f(t)-f(3)}{t - 3}&=\frac{(4t^{2}+t - 2)-37}{t - 3}\
&=\frac{4t^{2}+t-39}{t - 3}\
&=\frac{4t^{2}-12t+13t - 39}{t - 3}\
&=\frac{4t(t - 3)+13(t - 3)}{t - 3}\
&=\frac{(t - 3)(4t + 13)}{t - 3}\
&=4t+13\quad(t
eq3)
\end{align*}
$$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

Step2: Calculate \(f(3)\)

Step3: Calculate \(f(t)\)

Step4: Calculate the average rate of change

Answer: